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A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Principal-Agent Models and Moral Hazard
Christopher W. Miller
Department of MathematicsUniversity of California, Berkeley
April 11, 2014
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Background and Motivation
One major goal of the bottom-up approach to economics is tomodel the decision-making of individuals, as well as collections ofindividuals in markets.
In competitive markets with complete information, this isrelatively well-understood.
With asymmetric information, contracts between individualscan go wrong in many ways.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Background and Motivation
Complete information
Efficient equilibria by Welfare Theorems
Incomplete information
Adverse selection (e.g. lemon markets)Moral hazard (e.g. principal-agent problem)
”All happy families are alike; each unhappy family isunhappy in its own way.” - Leo Tolstoy
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Background and Motivation
Goals of this talk:
Set up a basic continuous-time principal-agent model,
Derive conditions under which moral hazard is irrelevant,
Cast the choice of a contract as a Hamilton-Jacobi equation.
References:
”A Continuous-Time Version of the Principal-AgentProblem”, Sannikov (2007)
”How to build stable relationships between people who lie andcheat”, Ekeland (2013)
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Table of contents
1 A Principal-Agent ModelA Simple ModelSome ExamplesProblem Statement
2 Incentive-Compatible ContractsContinuation ValueGirsanov TheoremKey Theorem
3 Optimal Contract EquationsControl FormulationHamilton-Jacobi Equation
4 Conclusion
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
A Simple Model
Consider simple contract between two parties. The first (theprincipal) hires the second (the agent) to work for him. The agentis in charge of a project which generates a revenue stream dXt forthe principal:
dXt = At dt + σ dZt . (1)
Here Zt is a standard Brownian motion with filtration FZt and At
is FZt -adapted and constrained to a compact set [0, a]. In
exchange for work, the principal pays Ct .
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
A Simple Model
Assume principal is risk-neutral and agent is risk-averse, withrespective utilities:
(principal) rE[∫ ∞
0e−rt (dXt − Ct dt)
](2)
(agent) rE[∫ ∞
0e−rt (u(Ct)− h(At)) dt
]. (3)
We assume that r , u, and h are known to both parties.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
A Simple Model
In this case, the principal would want the agent to work as hard aspossible. They would require At = a and compensate at c suchthat u(c) = h(A) + ε. This gives all the power to the principal.
Condition (Moral Hazard)
Let FXt be the filtration generated by Xt . We require that the
process Ct is FXt -adapted.
Intuitively, this means compensation C must be a function of pastoutputs X , but cannot depend explicitly upon the effort A.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
A Simple Model
The agent can always blame low revenue (dXt < 0) on badluck (dZt < 0) instead of low effort (dAt < 0).
The principal would like to make Ct = φ(At), but cannot. Hemust use the information of past values of Xt to determine Ct
(non-Markovian).
The principal would like to threaten punishment if the agent iscaught lying. We exclude this with a second condition.
Condition (Limited Liability)
The agent’s compensation Ct ≥ 0 for all t.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
Some Examples
Consider a principal owns a farm in the remote countryside andhires an agent to tend to the farm. What are some contracts hecan offer?
1 Ct = c (fixed salary)
2 Ct = aX+t for 0 ≤ a ≤ 1 (share-cropping)
3 Ct = (Xt − c)+ for c ≥ 0 (farming)
Share-cropping and farming are geographically and historicallywidespread. Fixed salary would lead to the agent having noincentive to work.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
Some Examples
Moral hazard and limited liability are fundamental to modernfinance industry:
1 Fund managers effort is not easily observable by clients; poorreturns may be blamed on the market (moral hazard)
2 Losses come out of the pocket of the client, not the manager(limited liability)
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
Problem Statement
Definition
A contract is a pair (Ct ,At), with Ct ≥ 0 and FXt -adapted, while
At is FZt -adapted.
Definition
A contract (Ct ,At) is incentive-compatible (IC) if the effortprocess At maximizes the agent’s total expected utility.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
A Simple ModelSome ExamplesProblem Statement
Problem Statement
Definition
A contract (Ct ,At) is individually-rational (IR) if both the principaland agent have:
(principal) rE[∫ ∞
0e−rt (dXt − Ct dt)
]≥ 0 (4)
(agent) rE[∫ ∞
0e−rt (u(Ct)− h(At)) dt
]≥ 0. (5)
Goal
We want to find a contract (Ct ,At), which is bothincentive-compatible and individually-rational, that maximizes theprincipal’s expected utility.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Continuation Value
Let (Ct ,At) be a contract. We want a characterization of whenthis contract is incentive-compatible. Consider the followingnatural process:
Definition
We define the continuation value the agent derives from followingthe contract (Ct ,At) at time t as:
Wt = rE[∫ ∞
te−r(s−t) (u(Cs)− h(As)) ds | FZ
t
]. (6)
It is useful to derive an SDE governing this process.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Continuation Value
Lemma
There exists a unique FZt -adapted process Yt such that:
dWt = r (Wt − u(Ct) + h(At)) dt + rσYt dZt (7)
= r (Wt − u(Ct) + h(At)− YtAt) dt + rYt dXt . (8)
One key observation is that both the agent and the principal canevolve Wt using (7) and (8) respectively.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Continuation Value
Proof
The idea is to re-write Wt such that a martingale shows up, thenuse the Martingale Representation Theorem. In particular:
Wt = rE[∫ ∞
te−r(s−t) (u(Cs)− h(As)) ds | FZ
t
]= rertE
[∫ ∞0
e−rs (u(Cs)− h(As)) ds | FZt
]−rert
∫ t
0e−rs (u(Cs)− h(As)) ds.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Continuation Value
Under some growth assumptions, the expectation is a continuousFZt -martingale, so there exists unique FZ
t -adapted process Yt suchthat:
e−rtWt = rW0 + r
∫ t
0σe−rsYs dZs − r
∫ t
0e−rs (u(Cs)− h(As)) ds.
By Ito’s lemma, we see:
−re−rtWt dt +e−rt dWt = rσe−rtYt dZt−re−rt (u(Ct)− h(At)) dt
=⇒ dWt = r (Wt + u(Ct)− h(At)) dt + rσYt dZt .
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Continuation Value
Next, note that because the agent is assumed to follow thecontract, we have:
Zt =1
σ
(Xt −
∫ t
0As ds
),
so we can replace the dZt in the previous SDE to get:
dWt = r (Wt + u(Ct)− h(At)− YtAt) dt + rYt dXt .
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Girsanov Theorem
In the proceeding, we need to consider an alternative effort A∗t .Formally, At and A∗t generate different probability laws for Xt . Wecall them PA and PA∗
respectively. Now we have: ZAt = 1
σ
(Xt −
∫ t0 As ds
)PA-Brownian motion
ZA∗t = 1
σ
(Xt −
∫ t0 A∗s ds
)PA∗
-Brownian motion
related by:
ZA∗t = ZA
t +
∫ t
0
As − A∗sσ
ds. (9)
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Girsanov Theorem
Theorem (Girsanov)
Let Zt be a P-Brownian motion and at an adapted process. Definea new process by:
Mt = exp
(∫ t
0as dZs −
1
2
∫ t
0a2s ds
).
If M is uniformly integrable, then a new measure Q, equivalent toP, may be defined by:
dQdP
= M∞.
Furthermore, we have:
Zt −∫ t
0as ds is a Q-Brownian motion.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Key Theorem
Theorem (Characterization of IC Contracts)
A contract (Ct ,At) is IC if and only if
YtAt − h(At) = maxa{aYt − h(a)} a.e. (10)
Proof
First, consider an arbitrary alternative strategy A∗t and the processcorresponding to the return from following A∗t then switching to At :
Vt = r
∫ t
0e−rs (u(Cs)− h(A∗s )) ds + e−rtWt .
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Key Theorem
Our goal is to show Vt is a PA∗-submartingale. By Ito’s lemma and
(7):
dVt = re−rt (u(Ct)− h(A∗t )−Wt) dt + e−rt dWt
= re−rt (u(Ct)− h(A∗t )−Wt) dt
+re−rt (Wt − u(Ct) + h(At)) dt
+rσe−rtYt dZAt
= re−rt (h(At)− h(A∗t )) dt + rσe−rtYt dZAt .
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Key Theorem
Next, by (9), we rewrite this as:
dVt = re−rt (h(At)− h(A∗t ) + A∗t − At) dt +rσe−rtYt dZA∗t . (11)
Then if At does not satisfy (10) almost everywhere, choose A∗tthat does. Then the PA∗
-drift of Vt is positive and non-zero on aset of positive measure, so it is a strict submartingale. Inparticular, there exists t large enough that:
EA∗[Vt ] > V0 = W0.
Therefore, we conclude if At does not satisfy (10), then thecontract is not IC.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Continuation ValueGirsanov TheoremKey Theorem
Key Theorem
Alternatively, suppose At does satisfy (10). Then for anyalternative strategy A∗t , the expected gain process Vt is asupermartingale by (11). It is easy to see it is bounded below, sothere exists V∞ such that:
W0 = V0 ≥ EA∗[V∞] .
So the expected payoff of A is at least as good as that of A∗.Therefore, the contract is IC.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Control FormulationHamilton-Jacobi Equation
Control Formulation
Now if we assume for simplicity that the maximum in (10) occursin the interior, we conclude:
Yt = h′(At).
The maximization of the principal’s expected profit over ICcontracts then becomes:
max(Ct ,At)
{rE[∫ ∞
0e−rt (dXt − Ct dt)
]}s.t. dWt = r (Wt − u(Ct) + h(At)) + σh′(At) dZt
dXt = At dt + σdZt
W0 = w0, X0 = x0.
C. Miller Principal-Agent Models and Moral Hazard
A Principal-Agent ModelIncentive-Compatible Contracts
Optimal Contract EquationsConclusion
Control FormulationHamilton-Jacobi Equation
Hamilton-Jacobi Equation
These notes will be completed for next week...
C. Miller Principal-Agent Models and Moral Hazard
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